Shutter speeds are pretty easy to understand - but why do apertures increase so weirdly? How come bigger numbers are smaller apertures? And how the hell are you meant to remember all of this?
Don't worry - it's less of a mystery than you'd think. Here's how it works.

Why does it matter?

It's sort of important to know what 'half the aperture' is of any given aperture, because if you're shooting in Manual mode (you're not still shooting in automatic modes, are you?!), you can use something called 'synonymous exposures'. That is; you can use two different sets of exposure settings, that let the same amount of light onto your light sensor.

In other words: if you shoot a photo at 1/100 second, f/4 and ISO 200, you would get exactly the same brightness in your photo if you halved the ISO and doubled the shutter speed (so 1/50 second and ISO 100) or the other way around (1/200 second shutter speed and ISO 400).

ISO and Shutter speeds

Before we dive into aperture-land, let's take a look at shutter speeds. As I hinted at before, they are both pretty intuitive: A 2 second shutter speed means that the shutter is open twice as long as a 1 second shutter speed. The same goes for 1/100 and 1/200 - the former is half as fast as the latter.

ISO deals with light sensitivity; in the old days, we'd talk about film speeds of a certain ISO. These days, it refers to the sensitivity of your light sensor (or rather, a multiplication factor done by your camera's processing chips). When you're shooting at ISO 100, your camera will use the light capture data as-is. At ISO 200, it'll take your light measurements and multiply them by 2 (because it used half the shutter speed in its exposure calculation). At ISO 400, it multiplies everything by 4 - and so on.

F-stops and aperture

Aperture is the one that always trips up my students, because it appears as if the F-stop scale doesn't make any sense. For one thing, f/2.8 is a pretty weird number, but how can f/2.8 be a larger aperture opening than f/4?

The aperture opening is measured in f-stops, which are, in fact, a fraction. Specifically, an aperture opening is a fraction of the focal length of your lens. So, if you have a 100mm lens set to f/4, what you are really saying is that the aperture opening in the lens is 1/4th of 100mm. Let's do the math: 1/4th of 100mm is 25mm - or about an inch.

This fraction is obviously the reason why f/4 is a larger aperture than f/8 - if you get a 1/4 (25%) of a cake, you're getting more cake than if you're getting 1/8 (12.5%) of a cake.

You may have seen some very posh f/1.0 lenses (Canon's version of the 50mm f/1.0 costs around US$4,500), and if you've been around in the photography world for long enough, you may even have come across a couple of 50mm f/.95. If you're immediately wondering how this is possible, simply go back to the math: The 50mm f/1.0 lens has a 50mm aperture opening. As such, the 50mm f/.95 isn't a physics-defying freak of nature - it simply has an aperture opening that's larger than its focal length; 52.6 mm, to be exact.

The F-stop scale

The scale of F-stops is a geometric sequence of numbers: the sequence of the powers of the square root of 2. Which sounds pretty mystical, but have a look at the table that's hovering to the right of this text; the actual square roots on the left-hand side of the table might look arcane, but you'll no doubt recognise the aperture f-stops as ones that your camera comes up with when you're taking photos.

So why did they choose this scale? Well, F-stops increase and decrease go up and down (inverse-) geometrically in powers of the square root of two because when the aperture diameter increases by the square root of two, the size of the area of the aperture (in other words: the amount of light that hits the film/CCD) is doubled. The reason for choosing to use the 'square root of two' scale, then, is that it it keeps the intervals equivalent to the doubling and halving of exposures. Nifty, eh?

Of course, the f-stop scale doesn't operate exclusively with whole numbers; look at the second table to see what happens when I take a selection of fractions and plug them into the square root formula. Your camera will operate either with thirds- or halves of a stop, so when we look at different fractions of five (5-and-a half, 5-and-a-third and 5-and-two-thirds), we'll get more f-stops you'll probably recognise (f/6.7, f/6.3 and f/7.1, respectively).

This way of calculating apertures is the reason why you very rarely see f/3.0 or f/7.0, for example - although I have seen them on some cameras from time to time, bizarrely, especially camera phone cameras, who seem to relish throwing photographic convention to the wind.

Where do variable aperture lenses fit in?

You may have seen some lenses that have something like "18-35mm f/3.5-4.5" stamped on the side of them. That means that when the lens is zoomed out fully, you can shoot with a f/3.5 aperture. However, when you zoom in, you can only use f/4.5.

In a way, that makes sense, let's do the math. At 18mm, f/3.5 is a 5.1mm opening. At 35mm, f/4.5 is a 7.7mm opening. Those are pretty similar; due to the way the lens is designed, it is able to have a bigger aperture opening when it is fully zoomed out; but because it has a longer focal length, the aperture is a smaller fraction of the lens length at full zoom than at full wide-angle.

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